Question

Two point dipoles of dipole moment $\overrightarrow{p_{1}}$ and $\overrightarrow{p_{2}}$ are at a distance $x$ from each other and $\overrightarrow{p_{1}} \left|\right| \overrightarrow{p_{2}}.$ The force between the dipoles is :

• A. $\frac{1}{4\pi\varepsilon_{0}} \frac{4p_{1}p_{2}}{x^{4}}$
• B. $\frac{1}{3\pi\varepsilon_{0}} \frac{8p_{1}p_{2}}{x^{3}}$
• C. $\frac{1}{4\pi\varepsilon_{0}} \frac{6p_{1}p_{2}}{x^{4}}$
• D. $\frac{1}{4\pi\varepsilon_{0}} \frac{8p_{1}p_{2}}{x^{4}}$

Answer 1

The Correct Option is B

To determine the force between two point dipoles $\overrightarrow{p_{1}}$ and $\overrightarrow{p_{2}}$ which are a distance $x$ apart and aligned parallelly, we use the following theoretical understanding:

The interaction force between two aligned dipoles in free space can be calculated using the formula for the force between them when dipoles are aligned:

1. The formula for the force $F$ between two parallel dipoles is derived from the potential energy interaction between the dipoles in classical electromagnetism. The force can be given by: $$ F = \frac{1}{3\pi\varepsilon_{0}} \frac{8p_{1}p_{2}}{x^{3}}, $$ where $p_{1}$ and $p_{2}$ are the magnitudes of the dipole moments, $x$ is the distance between them, and $\varepsilon_0$ is the permittivity of free space.
2. This expression comes from simplifying the electromagnetic interaction between the dipoles when using the dipole approximation where $x$ is much larger than the size of the dipole.

By comparing with the given options, the correct answer is:

$$ \frac{1}{3\pi\varepsilon_{0}} \frac{8p_{1}p_{2}}{x^{3}}. $$

This is the only option that correctly matches the derivation for parallel dipoles.